\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 77, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/77\hfil Periodic solutions of Hamiltonian systems] {Existence of periodic solutions for non-autonomous second-order \\ Hamiltonian systems} \author[Y. Wu, T. An \hfil EJDE-2013/77\hfilneg] {Yue Wu, Tianqing An} \address{Yue Wu \newline College of Science, Hohai University, Nanjing 210098, China} \email{wyue007@126.com} \address{Tianqing An \newline College of Science, Hohai University, Nanjing 210098, China} \email{antq@hhu.edu.cn} \thanks{Submitted January 11, 2013. Published March 19, 2013.} \subjclass[2000]{34C25, 58F05} \keywords{Periodic solution; Hamiltonian systems; critical point; \hfill\break\indent variational method} \begin{abstract} The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second order Hamiltonian systems. New results are obtained by using the least action principle and the minimax methods, without the so-called Ahmad-Lazer-Paul type condition. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction and main results} Consider the second-order Hamiltonian system $$\label{eq} \begin{gathered} \ddot u(t)=\nabla F\big(t,u(t)\big),\\ u(T)-u(0)=\dot{u}(T)-\dot{u}(0)=0, \end{gathered}$$ where $T>0$ and $F:[0,T]\times\mathbb{R}^{N}\to\mathbb{R}$ satisfies the following assumption: \begin{itemize} \item[(A)] $F(t,x)$ is measurable in $t$ for every $x\in\mathbb{R}^{N}$, continuously differentiable in $x$ for a.e. $t\in[0,T]$, and there exist $a\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $b\in \mathscr{L}^1 (0,T;\mathbb{R}^{+})$ such that $$|F(t,x)|\leq a(|x|)b(t),\quad |\nabla F(t,x)|\leq a(|x|)b(t)$$ for all $x \in \mathbb{R}^{N}$ and a.e. $t \in [0,T]$. \end{itemize} The corresponding functional $\varphi : H^1_T \to \mathbb{R}$, $$\varphi (u)=\frac{1}{2}\int^T_0|\dot{u}(t)|^2dt + \int_0^T F \big( t,u(t) \big)dt$$ is continuously differentiable and weakly lower semi-continuous on $H^1_T$ (see \cite{MW89}), where $H_T^1$ is the usual Sobolev space with the norm $$\|u\|=\Big[ \int^T_0|u(t)|^2dt+ \int_0^T |\dot{u}(t)|^2dt \Big]^{1/2}.$$ It is well know that the solutions of problem \eqref{eq} correspond to the critical points of $\varphi$. Problem \eqref{eq} has been extensively studied in the past thirty years; see for example the references in this article. Under some suitable solvability conditions, such as the coercivity condition (cf. \cite{Berg77}), the periodicity condition (cf. \cite{Mawhin81}), the convexity condition (cf. \cite{Mawhin87}), the subadditive condition (cf. \cite{Tang95}), the existence and multiplicity results are obtained. We note that in many contributions (for example, see \cite{An11,MaTang02,Tang98A, WuXP99,YangRG08,ZhangXY08,ZhaoFK04}), the following condition was assumed: $$\label{eq2} \lim_{|x|\to \infty}|x|^{-2\alpha}\int^T_0 F(t,x)dt=\infty \quad \text{or}\quad -\infty,$$ where $\alpha$ is a constant. In this article, instead of \eqref{eq2}, we discuss the existence of periodic solutions of \eqref{eq} under a weak condition that $\liminf_{|x| \to \infty }|x|^{-2\alpha}\int^T_0 F(t,x)dt$ or $\limsup_{|x| \to \infty }|x|^{-2\alpha}\int^T_0 F(t,x)dt$ has appropriate lower or upper bound. Our main results are as follows: \begin{theorem}\label{thm1} Suppose that $F(t,x)=F_1(t,x)+F_2(x)$, where $F_1$ and $F_2$ satisfy assumption {\rm{(A)}} and the following conditions: \begin{itemize} \item[(F1)] there exist $f,g \in \mathscr{L}^1(0,T;\mathbb{R}^{+})$ and $\gamma \in [0,1)$ such that $$|\nabla F_1(t,x)|\leq f(t)|x|^{\gamma}+g(t),$$ for all $x\in \mathbb{R}^{N}$ and a.e. $t \in [0,T]$; \item[(F2)] there exist constants $r >0$ and $\alpha \in [0,2)$ such that $$(\nabla F_2(x)- \nabla F_2(y),x-y )\geq -r|x-y|^{\alpha},$$ for all $x,y \in \mathbb{R}^{N}$; item[(F3)] $$\liminf_{|x|\to \infty}|x|^{-2\gamma}\int_0^T F(t,x)\,dt \geq \frac{T^2}{8\pi^2} \int_0^T f^2(t)\,dt.$$ \end{itemize} Then problem \eqref{eq} has at least one periodic solution which minimizes $\varphi$ on $H_T^1$. \end{theorem} \begin{theorem}\label{thm2} % Th 1.2 Suppose that $F(t,x)=F_1(t,x)+F_2(x)$, where $F_1$ and $F_2$ satisfy assumptions {\rm (A), (F1), (F2)} and the following conditions: \begin{itemize} \item[(F4)] there exist $\delta \in [0,2)$ and $C>0$ such that \begin{equation*} \left( \nabla F_2(x)-\nabla F_2(y),x-y \right ) \leq C|x-y|^{\delta}, \end{equation*} for all $x,y \in \mathbb{R}^N$; \item[(F5)] $$\limsup_{|x|\to \infty}|x|^{-2\gamma}\int_0^T F(t,x)\,dt \leq -\frac{3T^2}{8\pi^2}\int_0^T f^2(t)\,dt.$$ \end{itemize} Then problem \eqref{eq} has at least one periodic solution which minimizes $\varphi$ on $H_T^1$. \end{theorem} \begin{theorem}\label{thm3} % Th 1.3 Suppose that $F(t,x)=F_1(t,x)+F_2(x)$, where $F_1$ and $F_2$ satisfy assumptions {\rm (A), (F1)}, and the following conditions: \begin{itemize} \item[(F2')] there exists a constant $01/2$ and $0<\mu < 2\lambda^2$, such that \begin{equation*} \left( \nabla F_2(t,x),y \right ) \geq -k(t)G(x-y), \end{equation*} for all $x,y \in \mathbb{R}^N$ and a.e. $t\in [0,T]$; \item[(F7)] \begin{gather*} \limsup_{|x|\to \infty}|x|^{-2\gamma}\int_0^T F_1(t,x)\,dt \leq -\frac{3T^2}{8\pi^2}\int_0^T f^2(t)\,dt, \\ \limsup_{|x|\to\infty}|x|^{-\beta}\int_0^TF_2(t,x)\,dt \leq -8\mu \max_{|s|\leq 1}G(s)\int_0^T k(t)\,dt , \end{gather*} where $\beta=\log_{2\lambda}(2\mu)$. \end{itemize} Then problem \eqref{eq} has at least one periodic solution which minimizes $\varphi$ on $H_T^1$. \end{theorem} \begin{remark} \label{rmk1.5}\rm Theorems \ref{thm1}--\ref{thm3} extend some existing results. On the one hand, we decomposed the potential $F$ into $F_1$ and $F_2$. On the other hand, we weaken the so-called Ahmad-Lazer-Paul type condition \eqref{eq2} as conditions (F3), (F5) and (F3'). Note that \cite[Theorem 2]{YangRG08} and \cite[Theorem 1]{MaTang02} are the direct corollaries of Theorem \ref{thm1} and Theorem \ref{thm3} respectively. If $F_2=0$, \cite[Theorems 1 and 2]{TangXH10} are special cases of Theorem \ref{thm1} and Theorem \ref{thm2} respectively. Some examples of $F$ are given in section 3, which are not covered in the references. Moreover, our Theorem \ref{thm4} is a new result. \end{remark} \section{Proof of Theorems} For $u \in H_T^1$, let $\bar{u}=\frac{1}{T}\int_0^T u(t)\,dt ,\quad \tilde{u}(t)=u(t)-\bar{u}.$ The following inequalities are well known (cf. \cite{MW89}): \begin{gather*} \|\tilde{u}\|_{\infty}^2\leq \frac{T}{12}\|\dot u\|^2_{L^2} \quad\text{(Sobolev's inequality)},\\ \|\tilde{u} \|_{L^2}^2\leq \frac{T^2}{4\pi^2}\|\dot u\|^2_{L^2}\quad \text{(Wirtinger's inequality)} \end{gather*} For convenience, we denote $$M_1=\Big( \int_0^T f^2(t)\,dt \Big)^{1/2},\quad M_2=\int_0^T f(t)\,dt,\quad M_3=\int_0^T g(t)\,dt.$$ Now we give the proofs of the main results. \begin{proof}[Proof of Theorem \ref{thm1}] By (F3), we can choose an $a_1>T^2/(4\pi^2)$ such that $$\label{3.1} \liminf_{|x| \to \infty}|x|^{-2\gamma}\int_0^T F(t,x)\,dt>\frac{a_1}{2}M_1^2.$$ By (F1) and the Sobolev's inequality, for any $u \in H_T^1$, \label{3.2} \begin{aligned} &\big|\int_0^T[F_1(t,u(t))-F_1(t,\bar{u})]\,dt\big|\\ &=\big| \int_0^T{\int_0^1 \left(\nabla F_1 \big(t,\bar{u}+s\tilde{u}(t) \big),\tilde{u}(t)\right) \,ds}\,dt \big| \\ &\leq \int_0^T{\int_0^1 f(t) | \bar u + s \tilde {u}(t) |^{\gamma} |\tilde{u}(t)| \,ds}\,dt + \int_0^T{\int_0^1 g(t) |\tilde{u}(t)| \,ds}\,dt \\ &\leq |\bar u|^{\gamma} \Big(\int_0^T f^2(t)\,dt \Big)^{1/2} \Big(\int_0^T |\tilde u(t)|^2\,dt \Big)^{1/2}\\ &\quad + \|\tilde u\|^{\gamma +1}_{\infty} \int_0^T f(t)\,dt + \|\tilde u\|_{\infty} \int_0^T g(t)\,dt \\ &\leq \frac{1}{2{{a}_1}}\|\tilde{u}\|_{L^2}^2+ \frac{{{a}_1}}{2}M_1^2{{|\bar{u}|}^{2\gamma }} +{{M}_2}\|\tilde{u}\|_{\infty }^{\gamma +1}+{{M}_3} {{\|\tilde{u}\|}_{\infty }} \\ &\leq \frac{{{T}^2}}{8{{\pi }^2}{{a}_1}} \|\dot{u}\|_{L^2}^2 + \frac{{{a}_1}}{2}M_1^2{{ |\bar{u}|}^{2\gamma }} + {{\big(\frac{T}{12}\big) }^{ \frac{\gamma +1}{2}}} {{M}_2} \|\dot{u}\|_{L^2}^{\gamma +1} + {{\big(\frac{T}{12}\big) }^{1/2}} {{M}_3}{{\|\dot{u}\|}_{L^2}} \end{aligned} Similarly, by (F2) and the Sobolev's inequality, for any $u \in H_T^1$, \label{3.3} \begin{aligned} \int_0^T {[ {{F}_2}(u(t))-{{F}_2}(\bar{u})]dt} &= \int_0^T {\int_0^1 {\frac{1}{s}\left( \nabla {{F}_2} (\bar{u}+s\tilde{u}(t))-\nabla {{F}_2}(\bar{u}),s\tilde{u}(t) \right)}\,ds\,dt} \\ &\ge - \int_0^T {\int_0^1 rs^{\alpha -1}} {{\left| \tilde{u}(t) \right|}^{\alpha }}\,ds\,dt\\ &\ge -\frac{rT}{\alpha } \|\tilde{u}\|_{\infty }^{\alpha }\\ &\ge -\frac{rT}{\alpha }{{\big(\frac{T}{12}\big)}^{{\alpha }/{2}}} \|\dot{u}\|_{L^2}^{\alpha } \end{aligned} It follows from \eqref{3.2} and \eqref{3.3} that \begin{align*} \varphi (u) &=\frac{1}{2}\|\dot{u}\|_{L^2}^2 + \int_0^T {\left[ {{F}_1} (t,u(t))-{{F}_1}(t,\bar{u}) \right]dt}\\ &\quad + \int_0^T {\left[ {{F}_2}(u(t))-{{F}_2} (\bar{u}) \right]dt}+ \int_0^T {F(t,\bar{u})}dt \\ &\ge \Big( \frac{1}{2}-\frac{{{T}^2}}{8{{\pi }^2}{{a}_1}} \Big) \|\dot{u}\|_{L^2}^2-{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}} {{M}_2}\|\dot{u}\|_{L^2}^{\gamma +1}-{{\big(\frac{T}{12}\big)}^{1/2}} {{M}_3}{{\|\dot{u}\|}_{L^2}}\\ &\quad -\frac{rT}{\alpha }{{\big(\frac{T}{12}\big)}^{\alpha/2}} \|\dot{u}\|_{L^2}^{\alpha } +{{|\bar{u}|}^{2\gamma }} \Big( {{|\bar{u}|}^{-2\gamma }} \int_0^T {F(t,\bar{u})dt}-\frac{{{a}_1}}{2}M_1^2 \Big) \end{align*} for all $u \in H^1_T$, which implies that $\varphi (u) \to \infty$ as $\|u\| \to \infty$, due to \eqref{3.1} and $\gamma <1$. By the least action principle (see \cite[Theorem 1.1 and Corollary 1.1]{MW89}), the proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] \emph{Step 1.} We firstly show that $\varphi$ satisfies the (PS) condition. Suppose that $\{u_n\}$ is a (PS) sequence, that is, ${\varphi }'({{u}_n})\to 0$ as $n\to 0$ and $\left\{ \varphi \left( {{u}_n} \right) \right\}$ is bounded. By (F5), we can choose an ${{a}_2}>T^2/(4\pi^2)$ such that $$\label{3.4} \underset{|x|\to \infty }{\mathop{\lim \sup }}\,{{|x|} ^{-2\gamma }}\int_0^T{F\left( t,x \right)dt} <-\Big(\frac{{{a}_2}}{2}+ \frac{\sqrt{{{a}_2}}T}{2\pi } \Big)M_1^2.$$ In a way similar to the proof of Theorem \ref{thm1}, one has \begin{aligned} &\int_0^T {\left( \nabla{{F}_1}(t,{{u}_n}(t)),{{{\tilde{u}}}_n}(t) \right) \,dt}\\ &\le \frac{{{T}^2}}{8{{\pi }^2}{{a}_2}} \left\| {{{\dot{u}}}_n} \right\|_{L^2}^2+\frac{{{a}_2}} {2}M_1^2{{|\bar{u}_n|}^{2\gamma}} +{{\big(\frac{T}{12}\big)}^{\frac{( \gamma +1)}{2}}} {{M}_2}\left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\gamma+1} +{{\big(\frac{T}{12}\big)}^{1/2}}{{M}_3}{{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}} \end{aligned}\label{3.5} and $\int_0^T {\left( \nabla {{F}_2}({{u}_n}(t)),{{{\tilde{u}}}_n}(t) \right)dt} \ge-\frac{rT}{\alpha}{{\big(\frac{T}{12}\big)}^{{\alpha }/{2}}} \left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\alpha } \int_0^T{r(t)dt}$ for all $n$. Hence one has \begin{aligned} \left\| {{{\tilde{u}}}_n} \right\|&\ge \left( {\varphi }'({{u}_n}),{{{\tilde{u}}}_n} \right)\\ &=\left\| {{{\dot{u}}}_n}\right\|_{L^2}^2+\int_0^T {\left( \nabla F(t,{{u}_n}(t)),{{{\tilde{u}}}_n}(t) \right)}dt \\ & \ge \Big( 1-\frac{{{T}^2}}{8{{\pi }^2}{{a}_2}} \Big) \left\| {{{\dot{u}}}_n}\right\|_{L^2}^2-\frac{{{a}_2}}{2}M_1^2 {{\left| {{{\bar{u}}}_n}\right|}^{2\gamma }} -{{\big(\frac{T}{12}\big)}^{\frac{\left(\gamma +1 \right)}{2}}} {{M}_2} \left\|{{{\dot{u}}}_n} \right\|_{L^2}^{\gamma +1}\\ &\quad -{{\big(\frac{T}{12}\big)}^{1/2}}{{M}_3}{{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}} -\frac{rT}{\alpha}{{\big(\frac{T}{12}\big)}^{{\alpha}/{2}}} \left\|{{{\dot{u}}}_n}\right\|_{L^2}^{\alpha } \end{aligned}\label{3.6} for large $n$. It follows from Wirtinger's inequality that $$\label{3.7} \left\| {{{\tilde{u}}}_n} \right\|\le \frac{{{\left( {{T}^2}+4{{\pi }^2} \right)}^{{1}/{2}\;}}}{2\pi }{{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}}.$$ By \eqref{3.6} and \eqref{3.7}, \begin{aligned} \frac{{{a}_2}}{2}M_1^2{{|\bar{u}_n|}^{2\gamma }} &\ge \Big( 1-\frac{{{T}^2}}{8{{\pi }^2}{{a}_2}} \Big) \left\| {{{\dot{u}}}_n} \right\|_{L^2}^2-{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1 }{2}\;}} {{M}_2}\left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\gamma +1}-{{\big(\frac{T}{12}\big)}^{\frac{1}{2}\;}} {{M}_3}{{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}} \\ &\quad-\frac{rT}{\alpha}{{\big(\frac{T}{12}\big)}^{\frac{\alpha }{2}\;}} \left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\alpha } -\frac{{{\left( {{T}^2}+4{{\pi }^2} \right)}^{{1}/{2}\;}}}{2\pi } {{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}}\\ &\ge \frac{1}{2}\left\|{{{\dot{u}}}_n}\right\|_{L^2}^2+{{C}_1}, \end{aligned} \label{3.8} where \begin{align*} {{C}_1}=\min_{{s\in [0,+\infty)}}\Big\{ &\frac{4{{\pi}^2}{{a}_2}-{{T}^2}} {8{{\pi}^2}{{a}_2}}{{s}^2} -{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}}{{M}_2}{{s}^{{\gamma +1}}} -\big[\frac{rT}{\alpha}{{\big(\frac{T}{12}\big)}^{\alpha/2}}\big] {{s}^{\alpha}} \\ &\quad -\big[{{\big(\frac{T}{12}\big)}^{1/2}{{M}_3} +\frac{{{\big({{T}^2}+4{{\pi }^2} \big)}^{1/2}}}{2\pi }} \big]s\Big\}. \end{align*} Note that $a_2>T^2/(4\pi^2)$ implies $-\infty\frac{T^2}{4\pi^2-rT^2}$ such that $$\label{3.13} \liminf_{|x| \to \infty}|x|^{-2\gamma}\int_0^T F(t,x)\,dt>\frac{a_3}{2}M_1^2.$$ The condition (F2') and the Sobolev's inequality imply that \begin{align*} \int_0^T{\left[{{F}_2}(u(t)-{{F}_2}(\bar{u}))\right]dt} &=\int_0^T{\int_0^1{\frac{1}{s}\left(\nabla{{F}_2} (\bar{u}+s\tilde{u}(t))-\nabla {{F}_2}(\bar{u}),s\tilde{u}(t) \right)}\,ds\,dt} \\ &\ge-\int_0^T{\int_0^1{r}}s{{\left|\tilde{u}(t) \right|}^2}\,ds\,dt -\frac{{r{T}^2}}{8{{\pi }^2}}\|\dot{u}\|_{L^2}^2. \end{align*} It follows immediately from the similar method of the proof of Theorem \ref{thm1} that \begin{align*} \varphi (u) & =\frac{1}{2}\|\dot{u}\|_{L^2}^2+\int_0^T {F(t,u(t))}dt\\ & \ge \left( \frac{1}{2}-\frac{{{T}^2}}{8{{\pi }^2}{{a}_3}}-\frac{rT^2}{8\pi^2} \right) \|\dot{u}\|_{L^2}^2-{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}} {{M}_2}\|\dot{u}\|_{L^2}^{\gamma +1}\\ &{\quad} -{{\big(\frac{T}{12}\big)}^{1/2}} {{M}_3}{{\|\dot{u}\|}_{L^2}} +{{|\bar{u}|}^{2\gamma }} \Big({{|\bar{u}|}^{-2\gamma}}\int_0^T{F(t,\bar{u})dt}-\frac{{{a}_3}}{2}M_1^2 \Big) \end{align*} for all $u \in H^1_T$, which implies that $\varphi (u) \to \infty$ as $\|u\| \to \infty$ by \eqref{3.13}, due to the facts that $\gamma <1$, $r< \frac{4\pi^2}{T^2}$ and $\| u\| \to \infty$ if and only if ${{\big( {{|\bar{u}|}^2}+\|\dot{u}\|_{L^2}^2 \big)}^{1/2}}\to \infty.$ By the least action principle, Theorem \ref{thm3} holds. \end{proof} \begin{proof}[Proof of Theorem \ref{thm4}] We firstly show that $\varphi$ satisfies the (PS) condition. Suppose that $\{u_n\}$ satisfies ${\varphi }'({{u}_n})\to 0$ as $n\to 0$ and $\left\{ \varphi \left( {{u}_n} \right) \right\}$ is bounded. By (F7), we can choose an $a_4>T^2/(4\pi^2$ such that $$\label{3.14} \limsup_ {|x|\to \infty }{{|x|}^{-2\gamma }} \int_0^T {{{F}_1}\left( t,x \right)dt} <-\big(\frac{{{a}_4}}{2}+\frac{\sqrt{{{a}_4}}T}{2\pi } \big)M_1^2.$$ By the ($\lambda$,$\mu$)-subconvexity of $G(x)$, we have $$\label{3.15} G(x)\le \left( 2\mu {{|x|}^{\beta }}+1 \right){{G}_0}$$ for all $x \in \mathbb{R}^N$, and a.e. $t \in [0,T]$, where ${{G}_0}=\max_{|s|\le 1} G(s)$, $\beta ={{\log }_{2\lambda }}( 2\mu)<2$. Then \begin{aligned} \int_0^T{\left( \nabla {{F}_2}\left( t,{{u}_n}(t) \right),{{{\tilde{u}}}_n}(t) \right)dt} &\ge-\int_0^T{k(t)G({{{\bar{u}}}_n})dt}\\ &\ge -\int_0^T{k(t)\left( 2\mu {{|\bar{u}_n|}^{\beta }}+1 \right){{G}_0}dt}\\ &=-2\mu {{M}_4}{{|\bar{u}_n|}^{\beta }}-{{M}_4}, \end{aligned}\label{3.16} where ${{M}_4}={{G}_0}\int_0^T{k(t)dt}$. It follows from \eqref{3.5} and \eqref{3.16} that for large $n$, \begin{aligned} \left\| {{{\tilde{u}}}_n} \right\| &\ge \left( \varphi ({{u}_n}),{{{\tilde{u}}}_n} \right)\\ &=\left\| {{{\dot{u}}}_n} \right\|_{L^2}^2+\int_0^T{\left( \nabla F(t,{{u}_n}(t)),{{{\tilde{u}}}_n}(t) \right)}dt\\ & \ge \Big( 1-\frac{{{T}^2}}{8{{\pi }^2} {{a}_4}} \Big)\left\| {{{\dot{u}}}_n} \right\|_{L^2}^2-\frac{{{a}_4}}{2}M_1^2{{|\bar{u}_n|}^{2\gamma }}-{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}\;}}{{M}_2}\left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\gamma +1} \\ & \quad -{{\big(\frac{T}{12}\big)}^{{1}/{2}\;}}{{M}_3}{{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}} -2\mu {{M}_4}{{|\bar{u}_n|}^{\beta }}-{{M}_4}. \end{aligned}\label{3.17} Then \eqref{3.7} and \eqref{3.17} imply that \begin{aligned} \frac{{{a}_4}}{2}M_1^2{{|\bar{u}_n|}^{2\gamma }} +2\mu {{M}_4}{{|\bar{u}_n|}^{\beta }} &\ge \Big( 1-\frac{{{T}^2}}{8{{\pi}^2}{{a}_4}} \Big) \left\| {{{\dot{u}}}_n} \right\|_{L^2}^2-{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}} {{M}_2}\left\|{{{\dot{u}}}_n}\right\|_{L^2}^{\gamma +1} \\ & \quad -\Big( {{\big(\frac{T}{12}\big)}^{1/2}} {{M}_3}+\frac{{{\left( {{T}^2} +4{{\pi }^2} \right)}^{{1}/{2}\;}}}{2\pi } \Big){{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}}-{{M}_4}\\ & \ge \frac{1}{2}\left\| {{{\dot{u}}}_n} \right\|_{L^2}^2+{{C}_4}, \end{aligned} \label{3.18} where \begin{align*} {{C}_4}&=\min_{s\in[0,+\infty)}\Big\{ \frac{8{{\pi }^2}{{a}_4}-{{T}^2}}{8{{\pi }^2}{{a}_4}}{{s}^2} -{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}}{{M}_2}{{s}^{\gamma +1}}-{M}_4\\ &\quad -\Big[{{\big(\frac{T}{12}\big)}^{1/2}}{{M}_3} +\frac{{{\left( {{T}^2}+4{{\pi}^2} \right)}^{1/2}}}{2\pi } \Big]s\Big\}. \end{align*} Note that $-\infty < C_4<0$ due to $a_4>\frac{T^2}{4\pi^2}$. By \eqref{3.18}, one has $$\label{3.19} \left\| {{{\dot{u}}}_n} \right\|_{L^2}^2\le {{a}_4}M_1^2 {{|\bar{u}_n|}^{2\gamma }}+4\mu {{M}_4}{{|\bar{u}_n|}^{\beta }}-2{{C}_4},$$ and then $$\label{3.20} {{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}} \le \frac{\sqrt{2{{a}_4}}}{2}{{M}_1}{{|\bar{u}_n|}^{\gamma }} +\sqrt{2\mu {{M}_4}}{{|\bar{u}_n|}^{\beta/2}}+{{C}_{5}},$$ where $C_5>0$. It follows from (F6) and \eqref{3.15} that \begin{aligned} &\int_0^T{\left[ {{F}_2}\left( t,u(t) \right)-{{F}_2}\left( t,\bar{u} \right) \right]dt}\\ &=-\int_0^T{\int_0^1{\left( \nabla {{F}_2}\left( t,{{{\bar{u}}}_n} +s{{{\tilde{u}}}_n}(t) \right),-{{{\tilde{u}}}_n}(t) \right)}\,ds\,dt} \\ & \le \int_0^T{\int_0^1{k(t)G\left( {{{\bar{u}}}_n}+(s+1){{{\tilde{u}}}_n}(t) \right)\,ds}\,dt} \\ & \le \int_0^T{\int_0^1{k(t)\left( 2\mu {{\left| {{{\bar{u}}}_n} +(s+1){{{\tilde{u}}}_n}(t) \right|}^{\beta }}+1 \right){{G}_0}\,ds}\,dt}\\ & \le 4\mu \int_0^T{k(t)\left( {{|\bar{u}_n|}^{\beta }}+{{2}^{\beta }} {{\left| {{{\tilde{u}}}_n}(t) \right|}^{\beta }} \right){{G}_0}dt} +{{G}_0}\int_0^T {k(t)dt} \\ & \le {{2}^{\beta +2}}\mu {{M}_4}\left\| {{{\tilde{u}}}_n} \right\|_{\infty }^{\beta } +4\mu {{M}_4}{{|\bar{u}_n|}^{\beta }}+{{M}_4} \\ & \le {{\big(\frac{T}{12}\big)}^{\beta/2}} {{2}^{\beta +2}}\mu {{M}_4}\left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\beta } +4\mu {{M}_4}{{|\bar{u}_n|}^{\beta }}+{{M}_4} \end{aligned} \label{3.21} for all $u\in H_T^1$. By the boundedness of $\{\varphi (u_n)\}$ and the inequalities \eqref{3.11}, \eqref{3.19}-\eqref{3.21}, one has \begin{align*} {{C}_{6}} &\le \varphi ({{u}_n}) \\ & =\frac{1}{2}\left\|{{{\dot{u}}}_n}\right\|_{L^2}^2 +\int_0^T{\left[ {{F}_1}\left( t,{{u}_n}(t) \right)-{{F}_1} \left( t,{{{\bar{u}}}_n} \right) \right]dt}\\ &\quad +\int_0^T{\left[ {{F}_2}\left( t,{{u}_n}(t) \right) -{{F}_2}\left( t,{{{\bar{u}}}_n} \right) \right]dt} +\int_0^T{F(t,{{{\bar{u}}}_n})dt} \\ & \le \Big( \frac{1}{2}+\frac{T}{4\pi \sqrt{{{a}_4}}} \Big) \left\| {{{\dot{u}}}_n} \right\|_{L^2}^2 +\frac{\sqrt{{{a}_4}}T}{4\pi }M_1^2{{|\bar{u}_n|}^{2\gamma }} +{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}}{{M}_2} \left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\gamma +1} \\ &\quad + {{\big(\frac{T}{12}\big)}^{1/2}}{{M}_3} {{\left\| {{{\dot{u}}}_n} \right\|}_{L^2}} + {{\big(\frac{T}{12}\big)}^{\beta/2}}{{2}^{\beta+2}} \mu{{M}_4}\left\| {{{\dot{u}}}_n} \right\|_{L^2}^{\beta} + 4\mu {{M}_4}{{|\bar{u}_n|}^{\beta }} + {{M}_4} \\ &\quad + \int_0^T {F(t,{{{\bar{u}}}_n})dt} \\ &\le \Big( \frac{1}{2}+\frac{T}{4\pi \sqrt{{{a}_4}}} \Big) \left( {{a}_4}M_1^2 {{|\bar{u}_n|}^{2\gamma }} +4\mu {{M}_4}{{|\bar{u}_n|}^{\beta }}-2{{C}_4} \right) +\frac{\sqrt{{{a}_4}}T}{4\pi }M_1^2{{|\bar{u}_n|}^{2\gamma }} \\ &\quad +{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}} {{M}_2}{{\left( \sqrt{{a}_4} {{M}_1}{{|\bar{u}_n|}^{\gamma }} +2\sqrt{\mu {{M}_4}}{{|\bar{u}_n|}^{\beta/2}}+{{C}_{5}} \right)}^{\gamma +1}} \\ &\quad + {{\big(\frac{T}{12}\big)}^{1/2}} p{{M}_3}\Big( \sqrt{{a}_4}{{M}_1} {{|\bar{u}_n|}^{\gamma }} +2\sqrt{\mu {{M}_4}}{{|\bar{u}_n|}^{\beta/2}}+{{C}_{5}} \Big)\\ &\quad +{{\big(\frac{T}{12}\big)}^{\beta/2}}{{2}^{\beta +2}}\mu {{M}_4} {{\left(\sqrt{{a}_4}{{M}_1}{{|\bar{u}_n|}^{\gamma }} +2\sqrt{\mu {{M}_4}}{{\left|{{{\bar{u}}}_n}\right|}^{\beta/2}}+{{C}_{5}} \right)}^{\beta}}\\ &\quad +\mu {{M}_4}{{|\bar{u}_n|}^{\beta}}+{{M}_4} +\int_0^T {F(t,{{{\bar{u}}}_n})dt} \\ & \le \Big(\frac{{{a}_4}}{2}+\frac{\sqrt{{{a}_4}}T}{2\pi } \Big) M_1^2{{|\bar{u}_n|}^{2\gamma }} +\Big( 6+\frac{T}{\pi \sqrt{{{a}_4}}} \Big)\mu {{M}_4}{{|\bar{u}_n|}^{\beta }} -\Big( 1+\frac{T}{2\pi \sqrt{{{a}_4}}} \Big){{C}_4} \\ &\quad +{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}}{{M}_2} \Big( {{2}^{\gamma}}{{a}_4}^{\frac{\gamma+1}{2}} {{M}_1}^{\gamma+1}{{|\bar{u}_n|}^{\gamma (\gamma +1)}} +{{2}^{3\gamma+1}}\mu^{\frac{\gamma+1}{2}} {M}_4^{\frac{\gamma+1}{2}}{{|\bar{u}_n|}^{\frac{\beta (\gamma +1)}{2}}}\\ &\quad +{{2}^{2\gamma }}C_{5}^{\gamma +1} \Big) +{{\big(\frac{T}{12}\big)}^{ \frac{\beta }{2}}}{{2}^{\beta +2}}\mu {{M}_4} \Big({{2}^{\beta -1}}{{{a}_4}^{\frac{\beta}{2}}}{{M}_1}^{\beta}{{|\bar{u}_n|} ^{\gamma \beta }} +{{2}^{3\beta -2}}\mu^{\frac{\beta}{2}} {{M}_4} ^{\frac{\beta}{2}}{{|\bar{u}_n|}^{\frac{{{\beta }^2}}{2}}}\\ &\quad +{{2}^{2\left( \beta -1 \right)}}C_{5}^{\beta } \Big) +{{\left( \frac{T}{12}\right)}^{1/2}}{{M}_3}\left( {\sqrt{{{a}_4}}}{{M}_1}{{|\bar{u}_n|}^{\gamma }} +2\sqrt{\mu {{M}_4}}{{|\bar{u}_n|}^{\beta/2}}+{{C}_{5}} \right)\\ &\quad +{{M}_4} +\int_0^T {F(t,{{{\bar{u}}}_n})dt} \\ & ={{|\bar{u}_n|}^{2\gamma }}\Big[ {{|\bar{u}_n|}^{-2\gamma }} \int_0^T {{{F}_1}(t,{{{\bar{u}}}_n})dt} + \Big( \frac{{{a}_4}}{2} + \frac{\sqrt{{{a}_4}}T}{2\pi } \Big)M_1^2\\ &\quad +{{\big(\frac{T}{12}\big)}^{1/2}}{\sqrt{{{a}_4}}}{{M}_1}{{M}_3} {{|\bar{u}_n|}^{-\gamma }} + {{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}} {{2}^{\gamma -\frac{1}{2}}}\sqrt{{{a}_4}}{{M}_1}{{M}_2} {{|\bar{u}_n|}^{\gamma (\gamma -1)}} \\ &\quad +{{\big(\frac{T}{12}\big)} ^{\beta/2}}{{2}^{2\beta +1}} \mu {{{a}_4}^{\frac{\beta}{2}}} {{M}_1}^{\beta}{{M}_4}{{|\bar{u}_n|} ^{\gamma (\beta -2)}} \Big] \\ &\quad +{{|\bar{u}_n|}^{\beta }} \Big[ {{|\bar{u}_n|}^{-\beta }} \int_0^T {{{F}_2}(t,{{{\bar{u}}}_n})dt} + \Big( 6 +\frac{T}{\pi \sqrt{{{a}_4}}} \Big)\mu {{M}_4}\\ &\quad + {{\big(\frac{T}{12}\big)}^{\beta/2}}{{2}^{4\beta }}\mu^{\frac{\beta+2}{2}}{{M}_4}^{ \frac{\beta+2}{2}} {{|\bar{u}_n|}^{\frac{1}{2}{{\beta }^2}-2}} \\ &\quad +{{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}}{{M}_2}{{2}^{3\gamma+1 }}\mu^{\frac{\gamma+1}{2}}{M_2}{M_4}^{\frac{\gamma+1}{2}} {{|\bar{u}_n|}^{\frac{\beta (\gamma -1)}{2}}} +{{\big(\frac{T}{12}\big)}^{1/2}}2{{M}_3}\sqrt{\mu {{M}_4}} {{|\bar{u}_n|}^{-\beta/2}} \Big] \\ &\quad - \Big( 1 + \frac{T}{2\pi \sqrt{{{a}_4}}} \Big){{C}_4} + {{\big(\frac{T}{12}\big)}^{\frac{\gamma +1}{2}}} {{2}^{2\gamma }} {{M}_2}C_{5}^{\gamma +1} + {{\big(\frac{T}{12}\big)}^{1/2}} {{M}_3}{{C}_{5}}\\ &\quad + {{\big(\frac{T}{12}\big)}^{\beta/2}} {{2}^{3\beta }}\mu {{M}_4}C_{5}^{\beta } + {{M}_4} \end{align*} for large $n$. The above inequality and \eqref{3.14} imply that $\{|\bar u|\}$ is bounded. Hence $\{u_n\}$ is bounded by \eqref{3.19}. By using the usual method, the (PS) condition holds. Similar to the proof of Theorem \ref{thm2}, we can verify that functional satisfies the other conditions of the saddle point theorem. We omit the details. \end{proof} \section{Examples} In this section, we give some examples of $F$ to illustrate that our results are new. \begin{example} \label{examp3.1}\rm Let $F=F_1+F_2$, with \begin{gather*} F_1(t,x)=\sin\big( \frac{2\pi t}{T}\big)|x|^{7/4}+(0.6T-t)|x|^{3/2} +\left(h(t),x \right),\\ F_2(x)=C(x)-\frac{3r}{4}|x|^{4/3}, \end{gather*} where $h\in \mathscr{L}^1(0,T;\mathbb{R}^N)$, $r>0$, $C(x)=\frac{3r}{4}(|x_1|^4+|x_2|^{4/3}+\cdots +|x_N|^{4/3})$. \end{example} By Young's inequality, it is easy to see that \begin{align*} |\nabla F_1(t,x)| &\le \frac{7}{4}\Big| \sin\Big( \frac{2\pi t}{T}\Big) \Big||x|^{3/4}+\frac{3}{2}|0.6T-t||x|^{1/2}+|h(t)|\\ &\le \frac{7}{4}\Big(\Big| \sin\Big( \frac{2\pi t}{T}\Big) \Big|+\varepsilon \Big) |x|^{3/4}+\frac{T^3}{\varepsilon^2}+|h(t)| \end{align*} for all $x \in \mathbb{R}^N$ and a.e. $t \in [0,T]$, where $\varepsilon >0$. And $\left( \nabla F_2(x)-\nabla F_2(y),x-y \right)\ge -r|x-y|^{4/3}$ for all $x,y \in \mathbb{R}^N$. Thus, (F1), (F2) hold with $\gamma=3/4$, $\alpha=4/3$ and $$f(t)=\frac{7}{4}( | \sin( \frac{2\pi t}{T}) | + \varepsilon ), \quad g(t)= \frac{T^3}{\varepsilon^2}+|h(t)|.$$ However, $F$ does not satisfy \eqref{eq2}. In fact \begin{align*} & |x|^{-2\gamma}\int_0^T F(t,x)dt\\ &=|x|^{-3/2} \int_0^T \big[\sin\big( \frac{2\pi t}{T} \big) |x|^{7/4} +(0.6T-t)|x|^{3/2}+\big(C(x)-\frac{3r}{4}|x|^{4/3}\big) + (h(t),x)\big]dt\\ &=0.1T^2+\frac{T(C(x)-\frac{3r}{4}|x|^{4/3})}{|x|^{3/2}} +\Big( \int_0^T h(t)dt, |x|^{-3/2}x \Big) \end{align*} On the other hand, we have $$\frac{T^2}{8\pi^2}\int_0^T f^2(t)dt= \frac{49T^3}{128\pi^2} \Big( \frac{1}{2}+\frac{4\varepsilon}{\pi}+\varepsilon^2 \Big)$$ If $T<\frac{128\pi^2}{245}$, we choose $\varepsilon>0$ sufficient small such that $$\liminf_{|x|\to \infty}|x|^{-2\gamma}\int_0^T F(t,x)dt=0.1T^2 >\frac{T^2}{8\pi^2}\int_0^T f^2(t)dt$$ which implies that (F3) holds. Then $F=F_1+F_2$ is not convex, not $\gamma$-subadditive, not periodic, not a.e. uniformly coercive, and $\nabla F$ is not sublinear. Thus, $F$ is not covered by results in the references. \begin{example} \label{examp3.2}\rm Let $F=F_1+F_2$, with \begin{gather*} F_1(t,x)=(0.5T-t)|x|^{7/4}+(0.4T-t)|x|^{3/2}+\left(h(t),x\right),\\ F_2(x)=-\frac{4r}{5}|x|^{5/4}, \end{gather*} where $h\in \mathscr{L}^1(0,T;\mathbb{R}^N)$, $r>0$. \end{example} Similar to Example \ref{examp3.1}, we can see that all conditions of Theorem \ref{thm2} hold but $F$ is not covered by results in the references. \begin{example} \label{examp3.3}\rm Let $F=F_1+F_2$, with \begin{gather*} F_1(t,x)=(0.5T-t)|x|^{7/4}+(0.6T-t)|x|^{3/2}+\left(h(t),x\right),\\ F_2(x)=C(x)-\frac{r}{2}|x|^2, \end{gather*} where $h\in \mathscr{L}^1(0,T;\mathbb{R}^N)$, $C(x)=\frac{r}{2}(|x_1|^4+|x_2|^2+\cdots+|x_N|^2)$, $0 < r<\frac{4\pi^2}{T^2}$. \end{example} In a way similar to Example \ref{examp3.1}, it is easy to see that condition (F1) and (F2') are satisfied with $\gamma=3/4$. However, $F$ does not satisfies \eqref{eq2}. In fact, \begin{align*} &|x|^{-2\gamma}\int_0^T F(t,x)dt\\ &=|x|^{-2/3} \int_0^T \big[(0.5T-t)|x|^{7/4} +(0.6T -t)|x|^{3/2} + \left(C(x) - \frac{r}{2}|x|^2 \right)+ (h(t),x) \big]dt\\ &=0.1T^2+\frac{T\left( C(x)-\frac{r}{2}|x|^2\right)}{|x|^{3/2}} +\Big(\int_0^T h(t)dt,x|x|^{-3/2}\Big)\\ &=0.1T^2+\frac{rT(|x_1|^4-|x_1|^2)}{2|x|^{3/2}} +\Big(\int_0^T h(t)dt,x|x|^{-3/2}\Big). \end{align*} We can choose $\varepsilon>0$ small enough and some suitable $T$ such that \begin{align*} \liminf_{|x|\to\infty}|x|^{-2\gamma}\int_0^T F(t,x)dt=0.1T^2 >\frac{T^2}{2(4\pi^2-rT^2)}\int_0^T f^2(t,x)dt, \end{align*} which implies that (F3') holds. $F$ is also not covered by results in the references. \begin{thebibliography}{99} \bibitem{An11} Nurbek Aizmahin, Tianqing An; \emph{The existence of periodic solutions of non-autonomous second-order Hamiltonian systems.} Nonlinear Analysis TMA \textbf{74}, (2011), 4862--4867. \bibitem{Berg77} M. S. Berger, M. 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